Molecular Basis of Carbon Dioxide Transport in Polycarbonate

J. Phys. Chem. B 2008, 112, 4253-4260

4253

Molecular Basis of Carbon Dioxide Transport in Polycarbonate Membranes Leoncio Garrido,† Mar Lo´ pez-Gonza´ lez,† Enrique Saiz,‡ and Evaristo Riande*,† Instituto de Ciencia y Tecnologı´a de Polı´meros, CSIC, 28006 Madrid, Spain, and Departamento de Quı´mica Fı´sica, UniVersidad de Alcala´ , Alcala´ de Henares, Madrid, Spain ReceiVed: NoVember 21, 2007; In Final Form: January 18, 2008

Gas transport of carbon dioxide in poly[bisphenol A carbonate-co-4,4′-(3,3,5-trimethylcyclohexylidene)diphenol carbonate] films over a wide range of pressure is described. The interpretation of the experimental results in terms of the dual mode model allowed the evaluation of the parameters of the model that govern the gas permeation process. The value of the diffusion coefficient obtained for carbon dioxide at zero concentration was 2.4 × 10-8 cm2 s-1, at 303 K. This parameter was also measured by using pulsed field gradient NMR finding that its value reaches a nearly constant value of (2.7 ( 0.9) × 10-8 cm2 s-1, at 298 K, for diffusion times greater than 20 ms. Both the diffusion and solubility coefficients were also computed by using simulation methods based on the transition states theory and the Widom method, respectively. The value obtained for the diffusion coefficient was 1.8 × 10-8 cm2 s-1, at 303 K, which compares very favorably with the experimental measurements. The drop of the simulated solubility coefficient with increasing pressure is sharper than that of the experimental one, at low pressures, and similar, at high pressures.

Introduction Gas transport across polymeric membranes depends on the physical state of the polymer chains integrating the barriers. Owing to the fact that crystallites are opaque to gas transport, amorphous polymers are preferable for the preparation of barrier membranes for gases separation. Gas transport across membranes under the driving force of a unidirectional negative chemical potential involves three steps: solution of the gas in the membrane, its diffusion across the barrier and desorption of the gas at the other side of the membrane. Both the sorption and diffusion processes depend on the state, rubbery or glassy, of the membranes.1,2 In the case of rubbery membranes, the solubility coefficient can be determined from the variation of free energy involved in the solution process that can be calculated using the Flory-Huggins theory of polymer solutions. The theory predicts that at low pressures the solubility coefficient, S, obeys Henry behavior, but at high pressures S undergoes an anomalous increase.3,4 The predictions of the theory are in agreement with the experimental results reported for the solubility of gases in rubbery membranes.5,6 In the case of glassy membranes, the solubility coefficient decreases with increasing pressure reaching a constant value at moderately high pressures. This behavior is described by the dual mode model7,8 that assumes glassy membranes as formed by a continuous phase in which microvoids accounting for the excess volume are dispersed. Gas solution in the continuous phase of the membrane obeys Henry behavior whereas the microvoids behave as Langmuir sites where adsorption processes take place. Like in rubbery membranes, Henry’s solubility of gases in the continuous phase of glassy membranes can be estimated from the change in free energy involved in the gas solution process.9 * To whom correspondence should be addressed. Fax: 34-91-5644853. E-mail: [email protected]. † CSIC. ‡ Universidad de Alcala ´.

Micro-Brownian motions in rubbery membranes give rise to the formation of nonpermanent holes through which the molecules of diffusant jump. In the case of membranes in the glassy state, micro-Brownian motions are frozen and molecules of diffusant spend a relatively long time wandering in cavities until local fluctuations give rise to the formation of channels of suitable radii through which the diffusant can slide to a nearby cavity.10-12 Usually the diffusion coefficient in glassy and rubbery membranes is obtained from permeation measurements. Pulsed field gradient (PFG) NMR methods have been widely used to measure self-diffusion coefficients of fluids in bulk and confined geometries.13-15 Briefly, this nondestructive method of measuring molecular mobility consists of labeling molecules via the characteristic Larmor frequency of a given nucleus contained in them. It is achieved by overlying a well-defined magnetic field gradient over the sample volume and measuring the dephasing of the signal with, i.e., a spin echo type of rf pulse sequence, as shown by Stejskal et al.16 Generally, PFG NMR measurements in well-defined microporous systems are resonably understood.14,15 However, a glassy polymer membrane should be different from those systems since it lacks impermeable obstacles or barriers within. In fact, the self-diffusion coefficient of carbon dioxide across polyethylene membranes determined using the PFG NMR technique was 1 order of magnitude higher than that calculated from permeation measurements.17 Such a high discrepancy between the values of diffusion coefficients determined by microscopic and macroscopic techniques has also been observed in, i.e., the diffusion of propane in zeolites.18 Since the diffusion coefficient is obtained from the averages of the squares of the particles displacements, an increase of diffusion times ∆ would give a better account of the displacements of molecules of gas between cavities that presumably control the diffusion step in permeation measurements. To test this assumption, the diffusion coefficients of [13C]O2 in films of poly[bisphenol A carbonate-co-4,4′-(3,3,5trimethylcyclohexylidene) diphenol carbonate] (PTCDC), at a

10.1021/jp711080h CCC: $40.75 © 2008 American Chemical Society Published on Web 03/15/2008

4254 J. Phys. Chem. B, Vol. 112, No. 14, 2008 given concentration and in equilibrium conditions, were measured by 13C PFG NMR in a rather large diffusion times window. In this work attention is also paid to the simulation of trajectories of CO2 in virtual glassy PTCDC using the transition states approach (TSA).11,19 Once diffusive regime is achieved, the diffusion coefficient of the gas is determined. Diffusion coefficients obtained from both NMR experiments and simulations are compared with those experimentally determined from permeation measurements. Finally Monte Carlo techniques based on the Widom insertion method20 are used to simulate the sorption of CO2 in PTCDC as a function of pressure. Previous permeation studies have shown that membranes prepared from this polycarbonate exhibit larger permselectivity to gases than other alternative polycarbonates, without losing permeability. The aim of this work was to investigate the degree of concordance between the simulated transport results of CO2 in PTCDC membranes and those independently obtained by permeation and NMR techniques.

Garrido et al.

Figure 1. Variation of the pressure on the downstream chamber with time, at 303 K, for different upstream pressures: (0) 1.41, (b) 2.90, (4) 5.76, and (O) 12.12 bar. Continuous line corresponds to the pressure calculated by integration Fick’s second law.

Experimental Section Polycarbonate films of about 156 ( 6 µm thickness were prepared at 573 K by compression molding of pellets of PTCDC supplied by Aldrich. The time of residence of the film in the mold was 15 min and then the mold was rapidly cooled at room temperature. The glass transition temperature of the film was measured with a DSC7 Perking Elmer calorimeter at the heating rate of 10 K/min. By taking as Tg the temperature at which the endotherm obtained in the second scan departs from the baseline in the low-temperature region, the value of this quantity was 478 K. The density of the films determined by pycnometry was 1.21 g cm-3, at room temperature. Permeation measurements were performed in a cell made up of two compartments separated by the membrane, immersed in a water thermostat. The permeation area of the membrane was 3.46 cm2. After making vacuum in the two compartments of the permeation cell, carbon dioxide at a predetermined pressure was allowed to flow into one of the two compartments, named upstream chamber. Gas flowing from the upstream- to the downstream chamber of 49.17 cm3 of volume was monitored with a MKS 628/B transducer (10-4 to 1 mmHg) via a PC. The permeation experiments were carried out in isothermal conditions. Prior to each experiment, the air inlet was measured as a function of time and further subtracted from the curve pressure vs time recorded in the downstream chamber. Sorption measurements were carried out in isothermal conditions using an experimental device made up of two chambers separated by a valve. The sorption device was immersed in a thermostat bath set at the temperature of interest. Circular films 0.1 mm thick, separated by metallic grids to facilitate gas sorption, were introduced into one of the chambers, called sorption chamber. After vacuum in the two chambers was made, they were isolated from each other by closing the valve separating them. Then carbon dioxide at a given pressure was introduced into the chamber that acts as reservoir, and once thermal equilibrium was achieved, the gas was allowed to flow to the sorption chamber by suddenly opening and closing the valve separating them. The evolution of pressure with time in the gas sorption chamber was monitored with a Ruska model 7230 (0-35 bar) pressure sensor via a PC. To perform the NMR measurements, the films of PTCDC were cut in strips less than 800 µm wide and approximately 2.5 cm long and placed inside a thick-wall 10 mm o.d. NMR tube designed for NMR studies of pressurized gases. Prior to

Figure 2. Dependence of the concentration of carbon dioxide in the PTCDC film at several temperatures: (9) 293, (0) 303; (b) 313, and (O) 323 K. The curves joining the experimental results were obtained with eq 4 using the values of the dual mode parameters shown in Table 2.

filling the tube at a given pressure with [13C]O2, the air was removed by vacuum. The gas pressure was monitored with a transducer working in the range 0-10 bar. The measurements were performed in a Bruker Avance 400 spectrometer equipped with a 89 mm wide bore, 9.4 T superconducting magnet (13C Larmor frequency at 100.61 MHz). The reported data were acquired at 298 ( 1 K with a Bruker diffusion probe head using 90° 13C pulse lengths of about 13 µs. The diffusion time, ∆, was varied between 3 and 500 ms, using a pulsed field gradient spin echo sequence for short values of ∆, less than 15 ms, and a stimulated spin echo sequence for those >15 ms. The length of the gradient pulses, δ, varied between 0.45 and 2 ms. For each pair of δ and ∆ values, the amplitude of the gradient pulses varied stepwise (number of steps between 10 and 16) from 0 up to a maximum value of 23 T/m. The repetition rate was 10 s. Typically, 96 scans were averaged with a total acquisition time for these experiments of 4.3 h. In some cases, increased averaging lead to total acquisition times of 14 h. The apparent diffusion coefficient at a given ∆ was calculated by fitting the experimental data to the corresponding exponential function. Previously, the gradient was calibrated according to the spectrometer manufacturer protocol at 298 ( 1 K, using a sample of water doped with CuSO4 at 1 g/l and a value of the water diffusion coefficient equal to 2.3 × 10-5 cm2 s-1. Furthermore, the calibration was verified at the range of gradient values used experimentally by measuring the diffusion coef-

Carbon Dioxide Transport in PTCDC

J. Phys. Chem. B, Vol. 112, No. 14, 2008 4255

TABLE 1: Values of the Permeability, Diffusion and Apparent Solubility Coefficients of CO2 in the PTCDC Membrane, at 303 K p0, bar

P, barrers

108 × D, cm2 s-1

102 × S, cm3 cm-3(cmHg)-1

1.41 2.06 2.90 3.82 5.76 8.50 12.12 14.14

18.6 17.5 16.4 15.8 15.0 14.3 13.6 13.6

2.2 2.3 2.4 2.5 2.8 3.1 3.5 3.8

8.3 7.6 6.9 6.3 5.5 4.6 3.9 3.6

TABLE 2: Variation of the Values of kD, b, C′H, and S Obtained from the Pressure Dependence of the Concentration of Carbon Dioxide in the PTCDC Membranea T, K 288 303 318 333 a

102 × S, 102 × kD, cm3(STP) cm-3 103 × b, m3(STP) cm-3 C′H, (cm Hg)-1 (cmHg)-1 (cm Hg)-1 cm3(STP) cm-3 1.99 1.43 1.06 0.96

5.6 3.4 3.5 3.2

24.0 20.6 17.0 11.0

11.4 6.9 5.8 3.8

The values of S correspond to 1 bar.

ficient of dry glycerol. It was found a value of D ) 2.23 × 10-8 cm2 s-1, in good agreement with the results reported for this parameter elsewhere.21 Also, diffusion measurements for these two liquids were performed over a wide range of diffusion times to assess the stability of the gradients and whether artifacts due to eddy currents affect the measurements. In all cases, the repetition rate was 10 s.

Figure 3. Arrhenius plots for: (0) global solubility coefficient S at 1 bar, (O) Henry’s solubility constant kD, (b) gas concentration in Langmuir sites C′H and (2) gas-polymer affinity b.

In Table 1, the values of the permeability coefficient of carbon dioxide, at 303 K, for different upstream pressures are shown. The permeability coefficient decreases with increasing pressure in the region of low pressures, reaching a nearly constant value at moderate pressures. The diffusion coefficient undergoes a slight increase with pressure climbing from 2.2 × 10-8 cm2 s-1 at p0 ) 1.41 bar to 3.8 × 10-8 cm2 s-1 at p0 ) 14.14 bar. The concentration of carbon dioxide in the PTCDC membrane was measured at different pressures and temperatures, and the pertinent results are shown in Figure 2. As usual in glassy systems, the pressure dependence of the concentration, C, is described by the dual-mode model7,8

C ) k Dp +

Experimental Results (A) Permeation and Sorption. As usual, the isotherms showing the variation of the pressure, p, of carbon dioxide in the downstream chamber with time presents a transitory process at short times which converts into a steady-state process at long times where p is a linear function of time. Illustrative isotherms for different pressures of the upstream chamber, at 303 K, are shown in Figure 1. From the slope, dp/dt, of the isotherms in steady-state conditions, the permeation coefficient, P, is obtained as

{ }

dp(t) Vl P ) 3.59 lim p0AT tf∞ dt

(1)

where V is the volume of the downstream chamber, A is the permeation area, l is the thickness of the gas barrier, and p0 is the pressure of the gas in the upstream chamber. If V, l, and A are given in cgs units and the pressure in cmHg, the permeation coefficient is obtained in barrers [1 barrer ) 10-10 [cm3(STP) cm cm-2 s-1 (cm Hg)-1]. The diffusion coefficient, D, was determined by the method suggested by Barrer22

D ) l 2/6θ

(2)

where the time lag θ is the time at which the straight line defined by the plot p vs t in steady-state conditions intersects the abscissa axis. Usually D is given in cm2 s-1. The apparent solubility coefficient of the gas in the membranes was obtained from

S ) P/D

(3)

If P is given in barrers and D in cm2 s-1, S is given in cm3 (STP) cm-3 (cm Hg)-1 units.

bC′Hp 1 + bp

(4)

where kD is Henry’s constant, C′H is the gas concentration in Langmuir sites, and b represents an affinity gas-polymer parameter. Equation 4 describes the experimental results of Figure 2 using the fitting parameters presented in Table 2. The fitting parameters obey Arrhenius behavior as the plots of Figure 3 show. The values of the activation energies in kcal mol-1 for kD, b, C′H, and S ()C/p) are, respectively, -3.2, -2.1, -3.2, and -4.4. Accordingly, the solubility of CO2 in the polymer is an exothermic process. Thermodynamic arguments allow to relate Henry’s constant with the boiling temperature of the gas in the liquid state under 1 atm of pressure, Tb, its partial molar volume, V h , the polymer-gas (in the liquid state) enthalpic interaction, χ, and the latent heat of vaporization of the gas in the liquid state, λ. The pertinent expression is4,9

kD )

[

(

)]

Tb 22414 λ exp -(1 + χ) 176V h RTb T

(5)

where T is the working temperature and R the gases constant. The value obtained for χ is 1.3, a rather low value presumably arising from the fact that the quadrupole nature of carbon dioxide favors interactions with the polar polymer. According to the partial immobilized dual-mode model, the pressure dependence of the permeability coefficient is given by23

P ) kDDD +

C′HbDH FK ) kDDD 1 + 1 + bp 1 + bp

[

]

(6)

where DD is the diffusion of the gas in the continuous phase, F ) DH/DD is the mobile fraction of Langmuir species and K ) bC′H/kD. The plot of P against the reciprocal of 1/(1 + bp) at 303 K, shown in Figure 4, gives a reasonably good straight

4256 J. Phys. Chem. B, Vol. 112, No. 14, 2008

Garrido et al.

Figure 4. Fitting of eq 4 to the dependence of the permeability coefficient on pressure at 303 K. Notice that at this temperature the value of b obtained form sorption experiments is 3.4 × 10-3 (cm Hg)-1.

Figure 6. 13C NMR spectrum corresponding to a sample (strips) of PTCDC in the presence of gas [13C]O2. Two peaks associated to free, 124.2 ppm, and sorbed, 120.7 ppm, gas are observed suggesting the two pools of gas molecules are in slow exchange regime. The chemical shifts are referenced externally to TMS.

consecutive and appropriately spaced π/2 rf pulses generates an observable NMR signal (echo) centered at a time equal to 2τ (spin echo) or 2τ1 + τ2 (stimulated spin echo) from the first rf pulse, where τ1 is the time separation between the first two rf pulses and τ2 is the time elapsed between the second and the third rf pulses. The magnetic labeling is accomplished by applying two gradient pulses of amplitude and duration g and δ, respectively, spaced by a time ∆, the diffusion time. In the absence of motion, the loss of phase coherence of the NMR signal caused by the first gradient pulsed would be compensated by the second gradient pulse, but this would not be the case if molecular diffusion occurs during the time ∆. Then, an attenuation of the echo is observed as expressed by Figure 5. Scheme illustrating the pulse sequences used in the NMR experiments: (a) the standard Stejskal-Tanner pulsed field gradient spin echo pulse sequence and (b) the pulse field gradient stimulated spin echo pulse sequence.

line from whose intercept and slope values of 11.3 barrers and 9.6 barrers are obtained for kDDD and C′HbDH, respectively. These results show that the contribution of the Henry’s mode to the permeation is slightly larger than that of the Langmuir mode. (B) NMR Results. PFG NMR methods to measure molecular diffusion are based on the relationship between the resonance frequency of the nucleus of interest and the external magnetic field it experiences, as expressed by the Larmor equation: ω0 ) -γB0. The application of a magnetic field gradient across the sample volume labels magnetically molecules having NMR sensitive nuclei enabling the tracking of their motion over a given time, the diffusion time. In practice, this is accomplished using a spin echo type of pulse sequence, as first described by Stejskal et al.16 In the present work, PFG spin echo and stimulated spin echo sequences shown in Figure 5, panels a and b, respectively, were used. The application of π/2 radiofrequency (rf) pulse followed at τ by a π rf pulse or three

A(g) ) A(0) exp[-(bD)]

(7)

where A(g) and A(0) are the amplitude of the echo in the presence of a gradient pulse with amplitude g and 0, respectively, b ) (γgδ)2(∆ - δ/3) where γ is the gyromagnetic ratio of the nucleus being observed and D is the diffusion coefficient. Before performing the diffusion measurements, the 13C signal corresponding to [13C]O2 contained in the NMR tube with the polymer sample was analyzed. As illustrated in Figure 6, the 13C NMR spectrum exhibits two peaks at 124.2 and 120.7 ppm corresponding to the nonsorbed (free) and sorbed (in the polymer) gas fractions,24 respectively. The appearance of the 13C resonance at lower frequency, associated to the fraction of sorbed gas, is due to the interaction between the polymer chains and gas molecules which have a highly polarizable CdO bond. The fact that two peaks are observed is an indication about the presence of two populations of CO2 molecules in a slow exchange regime. Figure 7 illustrates the attenuation of the signal intensity corresponding to the sorbed [13C]O2 as a function of the g2 at a diffusion time of 120 ms. The PFG NMR data did not fit well to a monoexponential function with the exception of free [13C]O2. Since a continuum spectrum of diffusion coefficients

Carbon Dioxide Transport in PTCDC

J. Phys. Chem. B, Vol. 112, No. 14, 2008 4257

Figure 7. Plot of the peak intensity vs g2 corresponding to [13C]O2 sorbed in PTCDC obtained with a diffusion time, ∆, equal to 120 ms. The duration, δ, of the gradient pulse was kept constant to 2 ms and the amplitude of the field gradient, g, in gauss/cm, was varied between 115 and 2010. For each data point, 96 scans were averaged with a TR of 10 s.

TABLE 3: Diffusion Coefficients of [13C]O2 in PTCDC at 4.55 bar and 298 Ka δ, ms/∆, ms

108 × D, cm2 s-1

β

r2

1.00/2.60 1.00/4.00 1.00/4.00 2.00/6.00 2.00/6.00 2.00/20.00 1.00/60.00 2.00/60.00 1.00/120.00 2.00/120.00 2.00/240.00 2.00/500.00 0.74/2.43b

3900 (900) 206 (12) 178 (13) 26 (2) 31 (6) 1.7 (0.8) 1.4 (0.4) 3.4 (0.2) 2.4 (0.3) 3.5 (0.1) 3.3 (0.1) 3.3 (0.2) 990 000 (10 000)

1.14 (0,05) 0.77 (0.07) 0.70 (0.09) 0.7 (0.1) 0.6 (0.1) 0.5 (0.1) 0.41 (0.08) 0.72 (0.05) 0.55 (0.07) 0.84 (0.03) 0.99 (0.07) 0.88 (0.07) 0.94 (0.01)

0.998 0.982 0.988 0.967 0.966 0.975 0.970 0.991 0.978 0.998 0.993 0.994 0.999

Figure 8. Double logarithmic plot of D (cm2 s-1) as a function of ∆ (ms) for water (closed circles) and glycerin (open circles) and [13C]O2 in PTCDC (open squares) showing the reproducibility of the diffusion measurements for diffusion times >6 ms for all samples. The PFG NMR measurements in samples with high values of D (i.e., water, free gas) do not require the use of gradient pulses with large amplitudes and the impact of artifacts due to eddy currents is not significant. In these cases, as illustrated for water, shorter diffusion times could be used. For glycerin and [13C]O2 in PTCDC, some deviations attributed to eddy currents are observed at ∆ e 6 ms. The D values of carbon dioxide in polycarbonate reach a constant value of 2.7 × 10-8 cm2 s-1 when ∆ > 20 ms.

a Diffusion coefficients were obtained by performing a fractional exponential fit. s.d. in parenthesis. b Fit corresponding to free [13C]O2.

might be a more realistic description of the systems under consideration24 than that provided by a set of discrete values (i.e., multiexponential fit), the data were fitted to a fractional exponential

A(g) ) A(0) exp[-(bD)β]

(8)

where β is a “stretch” coefficient and the rest of the variables are previously described. The results of the fittings are shown in Table 3. It is observed that the value of the diffusion coefficient of [13C]O2 in PTCDC determined by NMR reaches a nearly constant value of (2.7 ( 0.9) × 10-8 cm2 s-1 for diffusion times greater than 20 ms, as illustrated in Figure 8. The corresponding values of β were found to vary between 0.41 and 0.99. (C) Diffusion and Sorption Simulation Results. The simulation of the diffusion step of carbon dioxide in the membrane was carried out using the TSA thoroughly described elsewhere.11,19,25-28 In brief, four H terminated oligomers, each one of them containing 10 repeating units such as the one shown in Figure 9 with x ) y ) 1 (i.e., 3288 atoms in total) were placed inside a box having Periodic Boundary Conditions (PBC) and a side length L ) 30 Å, as to reproduce the macroscopic density of the polymer membrane (ca. 1.21 g cm-3). The system was submitted to a series of simulated annealing and then allowed to equilibrate at the working temperature by means of the DL_POLY Molecular Dynamics package.29 Next, the PBC box containing the polymeric matrix was divided into 106 grid points obtained by dividing each side into 100 intervals. A

Figure 9. Scheme of the repeating unit of PTCDC.

molecule of CO2 was successively placed at each grid position and the interaction between the diffusant molecule and the atoms of the polymer was computed allowing for fluctuations of the polymer atoms with a root-mean-square value (customarily called smearing factor) ∆′ ) 0.3 Å and for different orientations of the diffusant molecule. The energies thus obtained were analyzed in order to determine the number, position and extent of the local minima (i.e., the positions at which the probability of finding one molecule of CO2 is a local maximum). Finally, the diffusion process was simulated by a random walk of the diffusant along the positions of minimum energy generated by Monte Carlo procedures. The average of the square of the displacements of carbon dioxide as function of time for 500 independently generated trajectories is represented in Figure 10 as a double logarithmic plot. Once diffusive regime is reached, that is, the slope of the plot of Figure 10 is 1, the diffusion coefficient is obtained by means of the Einstein relationship

{

1 ∂ D ) lim 〈[r(t) - r(t)0]2〉 6 tf∞ ∂t

}

(9)

Inspection of the data of Figure 10 shows that diffusive regime is reached at rather long times. Poor statistical sampling is responsible for the departure of the computed trajectories from diffusive regime at short times. The value obtained for D from

4258 J. Phys. Chem. B, Vol. 112, No. 14, 2008

Figure 10. Double logarithmic plot of the simulated mean-square displacements of the CO2 molecules vs time, at 303 K.

the results of Figure 9 is 1.8 × 10-8 cm2 s-1, which compares very favorably with the experimental results that lie in the range (2.2-3.5) × 10-8 cm2 s-1 for pressures ranging from 1.4 to 14.1 bar. The process of gas sorption in the membrane was simulated by a Widom method.25,28 In brief, Monte Carlo procedures were employed to perform attempts to insert and remove molecules of gas from the POC box containing the polymeric membrane according to probabilities computed by taking into account three factors (see eqs 11-24 in ref 25), namely (a) the number of gas molecules that will fill up the PBC box at fixed values of pressure and temperature assuming that the box contained no polymer at all, (b) the energetic interaction between inserted gas molecules and the polymer matrix actually contained in the box which were previously computed for the simulation of the diffusion process, and (c) the volume of the gas molecules which was employed to set up a hard spheres interaction potential among molecules of gas, i.e., overlapping among molecules inserted into the box were not allowed. Monte Carlo simulations consisted of 50 independently generated series of 3 × 106 cycles. Equilibrium between the insertions and removals of particles in the virtual matrix is reached in less than 106 cycles. The average of the particles inserted in the matrix, at equilibrium, were transformed into solubility coefficients, expressed in cm3 gas (STP) cm-3 (cm Hg)-1, and both the simulated solubility coefficients and the experimental ones are plotted as a function of pressure in Figure 11. As usual, the solubility coefficient decreases with increasing pressure but the drop in the simulated curve is larger than that observed in the experimental one. For pressures lying in the range 0 < p < 300 cm Hg, the simulated solubility coefficient is overestimated, whereas that value is underestimated for pressures higher than 300 cm Hg. In any case the agreement between simulated and experimental results is reasonable in the whole pressures range. Discussion According to the basic dual mode model, Langmuir modes are totally immobilized in such a way that the diffusion step occurs across the continuous phase of the membrane. In contrast, the partial immobilized model here used in the interpretation of gas permeation in glassy membranes, assumes that Langmuir mode species can partly be immobilized in glassy polymeric membranes. Several authors have also proposed a more detailed permeation model in which mutual exchanges among Henry

Garrido et al.

Figure 11. Variation of the simulated values of the solubility coefficient of CO2 in the PTCDC membrane with pressure.

and Langmuir modes are assumed.30,31 The analysis of the permeation results at the light of the partial immobilized mode shows that the term C′HbDH in eq 6 is 9.6 barrers, at 303 K. Since the fitting of eq 4 to the curve describing the pressure dependence of the concentration of gas in the membrane gives C′H ) 20.6 cm3 (STP) cm-3 and b ) 3.4 × 10-3 (cm Hg)-1, the value of DH in the partial immobilized model is 1.4 × 10-8 cm2 s-1, at 303 K. On the other hand, taking into account that kDDD ) 11.3 barrers and kD ) 1.43 × 10-2 cm3(STP) cm-3 (cm Hg)-1, the diffusion constant across the continuous phase of the glassy membrane, DD, reaches the value of 7.9 × 10-8 cm2 s-1 at 303 K. It is worthy to point out that DH is closer than DD to the value of the diffusion coefficient obtained from the time lag (2.2 × 10-8 cm2 s-1 at 1.41 atm). Moreover the mobile fraction of the Langmuir species is DH/DD = 0.18. The concentration of gas in the membrane is C ) CD + CH where, according to eq 4, CH ) C′Hbp/1 + bp and CD ) kDp. Assuming that the partial immobilized model holds, the flux of gas across the membranes in steady-state conditions is given by

J ) -DD

[

]

∂CD ∂CH ∂CD FK - DH ) -DD 1 + 2 ∂x ∂x (1 + bp) ∂x

(10)

where F and K are given in eq 6. In the development of eq 10, the equality ∂CH/∂x ) (1/kD)(∂CH/∂p)(∂CD/∂x) was used. Since CD ) C - CH, it is easy to show that eq 10 becomes

[ ] [ ]

FK (1 + bp)2 ∂C ∂C ) -D(C) J ) -DD ∂x ∂x K 1+ 2 (1 + bp)

(11)

FK (1 + bp)2 ∂C D(C) ) DD ∂x K 1+ 2 (1 + bp)

(12)

1+

where

1+

Equation 12 suggests that the diffusion coefficient of the gas a concentration zero is

D(0) ) DD

(11++FKK )

(13)

Carbon Dioxide Transport in PTCDC By substituting the values of DD, F, and K obtained from the experimental results into eq 13 one finds D(0) ) 2.4 × 10-8 cm2 s-1, in pretty good agreement with the results obtained by the time lag method in a wide range of pressures (see Table 1). These results suggest that diffusion coefficients obtained by the time lag method are closer to D(0) than to DD, the diffusion coefficient of carbon dioxide in the continuous membrane phase. It is worth noting that, contrary to what one would expect, the diffusion coefficient augments with increasing pressure. This at first sight anomalous behavior may be due to the plasticization effect of the carbon dioxide that facilitates the motion of the molecular segments. Plasticization increases with pressure and hence the increase of the apparent diffusion coefficient with pressure reflected in Table 1. Values of the diffusion coefficient of [13C]O2 in membranes measured by PFG NMR, reported in the literature, are more than 1 order of magnitude larger than those obtained by permeation results.17 Also the results for the diffusion coefficient obtained in this work undergo a sharp decrease with increasing ∆ in the region of short values of diffusion times. Above a certain value of ∆, the diffusion coefficient of carbon dioxide in the polymer matrix remains practically independent of the diffusion time. One can visualize the free volume in the glassy state made up of static free volume, essentially independent of the thermally accessible motions of polymer chains, corresponding to the microcavities. On the other hand, there exists the dynamic free volume derived from accessible conformational changes and segmental motions of the polymer chains that form channels through which a diffusant molecule slides from a cavity to a nearby one.32 The static and dynamic free volumes are related, respectively, to solubility and diffusivity processes.33 Obviously, the larger the size of the cavity where the molecules of CO2 are wandering, the closer D is to the free diffusion coefficient. The fact that mean square displacements in diffusive regime scales with the first power of time makes unlikely such a sharp decrease in the values of D, unless the system is microstructurally heterogeneous made up of wide variety of microcavities of different size. Another possibility is that eddy currents and gradient pulse instability are responsible for the rather sharp decrease of D with increasing ∆ in the short diffusion times region. To test this hypothesis, the self-diffusion coefficient of water and glycerine was measured at different diffusion times and the pertinent results are shown in Figure 8. It can be seen that the values of D for water are independent of ∆ for values of this parameter lying in the range 3-800 ms. However, the diffusion coefficient of glycerine undergoes a sharp decrease of D with increasing diffusion times in the region of short values of ∆, and remains constant for values of ∆ > 6 ms. This behavior seems to rule out that the decrease observed in the values of D for CO2 in the polymer reflects the selfdiffusion coefficient of the gas in microcavities of different size. The apparent diffusion coefficient obtained by PFG NMR measurements at long diffusion times appears to be independent of ∆ and in very good agreement with the values determined from permeation measurements, as well as with the results obtained by simulation of the sorption process. Large diffusion times provide the gas molecules a suitable window to probe their surroundings and the weighting of molecular interactions and barriers hindering diffusion.34,35 The stretched-exponential function has already been used to investigate the problem of molecular diffusion in systems with complex behavior due to their intrinsic heterogeneity,36 as it may be case here. However, it could be argued that, instead of assuming a continuous distribution of diffusion coefficients, a multicompartment model

J. Phys. Chem. B, Vol. 112, No. 14, 2008 4259 (i.e., a biexponential fit) would better describe the hydrodynamic behavior of CO2 in PTCDC. This would imply the assumption that, at least, two pools of CO2 molecules separated by some type of barrier, in slow or nonexchange regime, are present in the polymer matrix. Besides the lack of justification for presence impermeable or semipermeable barriers within the polymer matrix subject of study here, the biexponential fittings (data not shown) did not provide any meaningful insight to the observations made above. The results of fitting the PFG NMR data with the stretched-exponential function provide a set of β values that seems to increase with increasing diffusion times, but to confirm this observation more experiments are needed. The simulation of the diffusion coefficient using the TSA gives a good account of the diffusive step of CO2 in the glassy membrane. Similar good agreement between simulated diffusion coefficients and experimental results was found for lighter or less condensable gases such as oxygen, nitrogen and carbon monoxide, argon.26,27 TSA results are very sensitive to the smearing factor ∆′ whose value estimated by comparing TSA dynamics at short times with full dynamics lies in the range 0.3-0.4 Å. The information at hand suggests that values of ∆′ within a relatively short range allow to predict the diffusion of gases in glassy membranes with relatively good accuracy.25-28 Monte Carlo techniques based in the Widom method give a fairly good account of the sorption processes. In fact, simulated sorption curves displaying the pressure dependence of the concentration exhibit the same pattern as the same plots for experimental results. However, the values obtained for the solubility coefficients are only in reasonable agreement with the experimental results. In general, discrepancies between simulated and experimental results increase as pressure increases. Discrepancies are larger the larger is the gas condensability, presumably as a consequence of plasticization effects. Conclusions The closeness of the diffusion coefficient of carbon dioxide in PTCDC obtained from permeation experiments to that determined for this parameter in Langmuir sites suggests that gas mobility in this sites has a strong effect on the overall diffusion coefficient. NMR experiments show that, for long diffusion times, the molecules of carbon dioxide have a large window to probe their surroundings and the weighting of molecular interactions and barriers hindering diffusion increases. In this situation the apparent diffusion coefficient obtained comes close to the experimental result obtained from permeation measurements. The diffusion coefficient of carbon dioxide simulated using the TSA is in very good agreement with the diffusion coefficient measured by both permeation and NMR techniques. The simulated solubility coefficient is also in fair agreement with the experimental one in the whole pressures range. The rather good agreement between transport parameters obtained by three different techniques is a result that merits to be highlighted. Acknowledgment. One of the authors (L.G.) acknowledges the financial support provided by the “Fundacio´n Domingo Martı´nez.” This work was supported by Comunidad de Madrid (CAM Project: S-0505/MAT/0227) and CICYT (Projects: CTQ2005-04710/BQU, MAT2005-05648-C02-01). References and Notes (1) Kesting, R. E.; Fritzsche, A. K. Polymeric Gas Separation Membranes; Wiley-Interscience: New York, 1993; Chapter 1.

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